Isospectral surfaces of genus two and three (Q2898405)

Two Riemannian manifolds are said to be \textit{isospectral} if they have the same eigenvalues of the Laplace operator. The literature contains a diverse collection of pairs of isospectral non-isometric manifolds satisfying various geometric constraints; the paper under review focuses on surfaces of low genus. In genus two, the authors construct the following examples: isospectral non-isometric surfaces with variable curvature; isospectral potentials on a surface of constant curvature \(-1\); and isospectral but non-isometric Riemannian orbifolds. In genus three, they obtain examples of isospectral non-isometric surfaces with variable curvature. All the examples consist of infinite families, and all are constructed by gluing copies of a basic building block. Isospectrality is proved via Sunada's theorem. The authors give concrete descriptions of the groups used in Sunada's theorem, which allows the reader to check their claims directly.